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Properties

Label 2-414-9.7-c1-0-10
Degree $2$
Conductor $414$
Sign $0.805 - 0.592i$
Analytic cond. $3.30580$
Root an. cond. $1.81818$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (1.07 + 1.36i)3-s + (−0.499 − 0.866i)4-s + (−1.08 − 1.87i)5-s + (−1.71 + 0.246i)6-s + (1.41 − 2.44i)7-s + 0.999·8-s + (−0.708 + 2.91i)9-s + 2.16·10-s + (2.01 − 3.48i)11-s + (0.643 − 1.60i)12-s + (3.22 + 5.59i)13-s + (1.41 + 2.44i)14-s + (1.39 − 3.48i)15-s + (−0.5 + 0.866i)16-s + 1.41·17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.618 + 0.786i)3-s + (−0.249 − 0.433i)4-s + (−0.485 − 0.840i)5-s + (−0.699 + 0.100i)6-s + (0.533 − 0.923i)7-s + 0.353·8-s + (−0.236 + 0.971i)9-s + 0.686·10-s + (0.606 − 1.05i)11-s + (0.185 − 0.464i)12-s + (0.895 + 1.55i)13-s + (0.377 + 0.653i)14-s + (0.360 − 0.900i)15-s + (−0.125 + 0.216i)16-s + 0.343·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $0.805 - 0.592i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ 0.805 - 0.592i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35275 + 0.444097i\)
\(L(\frac12)\) \(\approx\) \(1.35275 + 0.444097i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-1.07 - 1.36i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (1.08 + 1.87i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.41 + 2.44i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.01 + 3.48i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.22 - 5.59i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.41T + 17T^{2} \)
19 \( 1 - 4.95T + 19T^{2} \)
29 \( 1 + (-0.341 + 0.592i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.11 - 1.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.39T + 37T^{2} \)
41 \( 1 + (3.85 + 6.67i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.14 + 1.98i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.787 - 1.36i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 5.20T + 53T^{2} \)
59 \( 1 + (-6.09 - 10.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.69 + 8.12i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.17 - 7.23i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 + 0.0997T + 73T^{2} \)
79 \( 1 + (-3.08 + 5.34i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.53 - 2.66i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 7.82T + 89T^{2} \)
97 \( 1 + (6.48 - 11.2i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.17154021534781490649752192113, −10.27080253748460771711481033961, −9.115821401262600645927463311606, −8.685214890325385966586283737085, −7.87987854803428482674730691267, −6.80608928411675164033441583496, −5.40537598049167697462748519969, −4.35022793925545223922576179711, −3.66591092797131336060302252929, −1.27229357327223662075674808017, 1.46438208488221911242876761172, 2.81996177160334691251047378060, 3.60274869710491767694107889271, 5.38206589417650481377440603701, 6.70763132681834816310265194180, 7.69021169940941618675694412700, 8.267304712041860339118877129280, 9.280956212036189020865859976939, 10.20438960075082325915218633844, 11.34717880349467860697455669645

Graph of the $Z$-function along the critical line